STATISTICS FOR A SIMPLE TELEPHONE EXCHANGE MODEL 957 



ities of mutually exclusive events, we obtain the following infinitesimal 

 relations for Fn{z, t): 



F„(z, t -\- At) = y(n + 1) / F„+i(2 - u, t) du 



+ a / F„-i{z - ?/, /) du + F„(z - riAt, t) 



J(n-1)A( 



•[1 — A((yn -\- a)] + oiAf), for any n. 



Expanding the penultimate term of the right side in powers of nAt, 

 and the left side in powers of At, we divide by At, and take the limit as 

 At approaches 0. Now 



lim — / F„+iiz - u, t) du = F^+i(z, t). 



At->0 At J r,M 



Thus, omitting functional dependence on z and / for convenience, we 

 reach the following partial differential eciuations for F,Xz, t): 



iF„ = ,(„ + l)F.,,, + aF,._,-„|F. ^^^^^ 



— [yn + a]Fn , for anj^ n. 



Since Z(0) = 0, we impose the following boundarj^ conditions: 



Fn{0, 0=0 for n > and t > 0, 



Fniz, 0) = p„ for z ^ 0, (10.2) 



Fn{z, 0) = for z <0, 



where the sequence }p„} forais an arbitrary iV(0) distribution that is 

 zero for negative n. 



To transform the equations, we introduce the Laplace-Stieltjes in- 

 tegrals 



<Pn{t, t) = I c-^' dF„{z, t), t ^0, Re (r) > 0, 

 Jo- 

 in which the Stieltjes integration is understood always to be on the 

 variable z. We note that 



' " 1 



/ e~^'Fn{z,t) dz = -<^„(f, 0, 

 Jo- C 



and that 



^«(r, = F„{0,t) + / e-'=^F,. {z,t) 



Jo dz 



dz. 



